Let's call the height of the building x.
Since the tree is growing halfway between the stake and the building, the distance from the stake to the tree is also x.
Therefore, the total distance from the stake to the building is 2x.
The tree is 12 feet tall, so the total distance from the stake to the top of the building is 2x + 12.
Since the rope is stretched between the top edge of the building and the stake, the length of the rope is also 2x + 12.
However, the length of the rope is also the hypotenuse of a right triangle formed by the building height, the distance from the stake to the tree (which is x), and the tree height (which is 12).
Using the Pythagorean theorem, we can set up the equation:
x^2 + 12^2 = (2x + 12)^2
Expanding and simplifying, we have:
x^2 + 144 = 4x^2 + 48x + 144
Subtracting 144 from both sides, we get:
x^2 = 4x^2 + 48x
Subtracting 4x^2 and 48x from both sides, we get:
0 = 3x^2 + 48x
Dividing both sides by 3, we get:
0 = x^2 + 16x
Factoring out an x, we have:
x(x + 16) = 0
Setting each factor equal to zero, we have two possible solutions:
x = 0 or x + 16 = 0
Since the height of the building cannot be zero, we ignore the solution x = 0.
Therefore, x = -16 is the only valid solution. However, since the height of a building cannot be negative, we ignore this solution as well.
Therefore, there is no valid solution for the height of the building.
A length of rope is stretched between the top edge of a building and a stake in the ground. The head of the stake is at ground level. The rope also touches a tree that is growing halfway between the stake and the building. If the tree is 12 feet tall, how tall is the building? (1 point)
8 ft
24 ft
16 ft
32 ft
1 answer
1 answer